Download Optical spectroscopy and electronic structure of the face-centered icosahedral quasicrystals Zn- R

Optical spectroscopy and electronic structure of
the face-centered icosahedral quasicrystals ZnMg-R (R=Y, Ho, Er)
V Karpus, S Tumenas, A Suchodolskis, Hans Arwin and W Assmus
Linköping University Post Print
N.B.: When citing this work, cite the original article.
Original Publication:
V Karpus, S Tumenas, A Suchodolskis, Hans Arwin and W Assmus, Optical spectroscopy
and electronic structure of the face-centered icosahedral quasicrystals Zn-Mg-R (R=Y, Ho,
Er), 2013, Physical Review B. Condensed Matter and Materials Physics, (88), 9.
http://dx.doi.org/10.1103/PhysRevB.88.094201
Copyright: American Physical Society
http://www.aps.org/
Postprint available at: Linköping University Electronic Press
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-98664
PHYSICAL REVIEW B 88, 094201 (2013)
Optical spectroscopy and electronic structure of the face-centered icosahedral quasicrystals
Zn-Mg-R (R = Y, Ho, Er)
V. Karpus,1,* S. Tumėnas,1 A. Suchodolskis,1 H. Arwin,2 and W. Assmus3
1
Center for Physical Sciences and Technology, Semiconductor Physics Institute, A. Goštauto 11, LT-01108 Vilnius, Lithuania
2
Laboratory of Applied Optics, Department of Physics, Chemistry and Biology, Linköping University, SE-58183 Linköping, Sweden
3
J. W. Goethe-Universität, Physikalisches Institut, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany
(Received 2 July 2013; revised manuscript received 27 August 2013; published 9 September 2013)
Results of optical spectroscopy studies of the face-centered icosahedral (fci) single-grain Zn-Mg-Y, Zn-Mg-Ho,
and Zn-Mg-Er quasicrystals (QCs) are presented. The dielectric function of the QCs was measured in the
0.01–6 eV spectral range by IR-UV spectroscopic ellipsometry and far infrared reflection spectroscopy techniques.
A theoretical scheme of optical conductivity calculations is extended to account for the Fermi level positions within
and below a pseudogap. The model of the QC electron energy spectrum, based on a band structure hypothesis,
is suggested, which treats the electronic subsystem as a nearly free electron gas affected by intersections of the
Fermi surface with several families of Bragg planes. The experimental optical spectra are reproduced in detail
by theoretical calculations carried out within the framework of the model. The parameters of the electron energy
spectrum deduced from an analysis of optical data are close to those previously determined in an analysis of fci
Zn-Mg-R valence band photoemission spectra.
DOI: 10.1103/PhysRevB.88.094201
PACS number(s): 71.23.Ft
I. INTRODUCTION
Investigations of the optical response of quasicrystals
(QCs) is one of the principal tools for determination of their
electronic structure. Optical studies of icosahedral Al-Mn,1
Al-Cu-Fe,2 Al-Cu-Fe-B,3 Al-Pd-Mn,3–5 Al-Pd-Re,6–8 and AlMn-Si9 quasicrystals reveal the main features of their optical
spectra to be the broad absorption peak at about 1–2 eV and
a strong suppression of the usual metallic Drude-type optical
transitions. Investigations of decagonal quasicrystalline AlCo-Cu,10 Al-Co-Cu-Si,10 and Al-Ni-Co11 phases show an
essential reduction of the Drude-type response in the plane
of quasicrystalline atomic arrangement as compared to the
optical response in the periodic, crystalline direction.
Our previous preliminary spectroscopic ellipsometry
studies12–14 of icosahedral Zn-Mg-R quasicrystals (where
R = Y, Ho, Er), as well as reflection spectroscopy studies
of Zn-Mg-Y and Zn-Mg-Tb by Chernikov et al.,15 revealed
a strong interband optical feature at about 1 eV, but a larger
Drude peak as compared to that in Al-based quasicrystals.
A distinctive feature of the Zn-Mg-R quasicrystals with
respect to most other QCs is the fact that their valence bands
are due predominantly to the sp-type electron states, and from
this point of view they are closer to simple metals. This can
be one of the reasons for the comparatively higher Drude
contribution in Zn-Mg-R.
The sp character of the Zn-Mg-R valence bands was
revealed by their photoemission (PE) studies.16–18 On the
basis of the PE data analysis, a model for the Zn-Mg-R
electron energy spectrum was suggested,16 which allowed for
a successful description of the experimental Zn-Mg-R valence
band PE spectra. The model is based on a band structure
hypothesis and treats the Zn-Mg-R electronic subsystem
as a nearly free-electron (NFE) gas, affected by a weak
quasicrystalline potential. The hypothetical Zn-Mg-R electron
energy spectrum in the vicinity of the Fermi level is determined
by the Fermi surface intersections with several families of
Bragg planes.
1098-0121/2013/88(9)/094201(12)
The suggested model for the electronic structure of Zn-MgR was applied to the analysis of optical spectra of the simple
icosahedral Zn-Mg-Ho quasicrystal19 and the face-centered
icosahedral (fci) Zn-Mg-R (R = Y, Ho, Er) QCs.20 The
theoretical optical spectra, simulated within the framework of
the model, qualitatively agreed with the experimental spectra.
Here we will present (Sec. II) new experimental data
for the optical response of Zn-Mg-R recorded by IR-UV
spectroscopic ellipsometry and FIR reflection spectroscopy
techniques, which allowed us to expand the spectral range of
Zn-Mg-R dielectric function and optical conductivity from
the previous window19,20 of 0.1–6 eV to the 0.01–6 eV
range. Additionally, we have employed a surface-preparation
plasma-etching technique, which leads to an increase in the
optical features in the new IR-UV data compared to those
recorded previously.
The NFE-model of the Zn-Mg-R electron energy spectrum
will be further developed (Sec. III). The algorithm of the
optical conductivity calculations, worked out by Ashcroft and
Sturm,21 will be extended (Sec. IV) to account for various
positions of the Fermi level with respect to a pseudogap, appropriate for the Zn-Mg-R quasicrystals under consideration.
Finally (Sec. V), the suggested electron energy spectrum
model will be applied for a detailed analysis of the experimental Zn-Mg-R optical conductivity spectra. The spectra can
be nicely explained within the framework of the theoretical
model. The parameters of the electron energy spectrum,
deduced from the optical data analysis, are in close agreement
with those obtained in an analysis of Zn-Mg-R photoemission
spectroscopy data.
II. EXPERIMENT
The fci single-grain Zn62 Mg29 Y9 , Zn65 Mg25 Ho10 , and
Zn65 Mg24 Er11 quasicrystals were grown by the liquidencapsulated top-seeded solution-growth method.22,23 The
x-ray diffraction patterns of the QCs consist of sharp, instrumental resolution limited Bragg peaks (Fig. 1) indicating a
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V. KARPUS et al.
PHYSICAL REVIEW B 88, 094201 (2013)
e1
(221001)
(332002)
fci Zn-Mg-Ho
2.30
2.34
(333101)
(422211)
(222100)
(311111)
(211111)
2.19
1.59
½g = 1.17
_
e4
1.53
½(333111)
intensity (arb. u.)
e3
e2
optical conductivity (Ω-1cm-1)
e6
1.35
1.28
e5
8000
Re σ(ω)
6000
4000
fci Zn-Mg-Ho
2000
Im σ(ω )
0
0.1
1
10
photon energy (eV)
50
60
FIG. 2. Influence of the Ar+ etching on the Zn-Mg-Ho optical
conductivity spectrum. Dashed and solid curves present data recorded
before and after sputtering, respectively.
70
2θ (deg)
FIG. 1. (Color online) X-ray powder diffractogram of a Zn-MgHo quasicrystal. Indices of the diffraction
peaks correspond to indices
of the reciprocal lattice vectors g = πa i mi ei . Numbers at the peaks
indicate half moduli of the g vectors in 108 cm−1 units. Dashed line
corresponds to the Fermi wave vector (Table I).
high structural quality of the quasicrystals. The mass density
of the QCs was measured by the Micrometrics AccuPyc
1330 setup. The valence electron concentrations, determined
from the recorded values, are presented in Table I.
The optical surfaces of single-grain QC samples were
prepared by a careful mechanical polishing as described in our
previous work.20 Additionally, as a final surface-preparation
procedure, we employed a plasma etching technique by Ar+
ion sputtering in UHV chamber. Figure 2 illustrates the
effect of the final plasma etching on the Zn-Mg-Ho optical
conductivity spectrum. The etching allowed for an increase of
Zn-Mg-R optical features by about 10%.
The IR-UV dielectric function ε(ω) of the quasicrystals
was measured with spectroscopic ellipsometry (SE). In the
spectral range 0.73–6 eV, the measurements were carried out
with a dual rotating compensators ellipsometer (RC2, J. A.
Woollam Co., Inc.) in steps of 1 nm at 50, 60, and 70 degrees
of the incident angle. In the IR spectral region 333–5000 cm−1 ,
a rotating-compensator Fourier-transform ellipsometer (IRSE,
J. A. Woollam Co., Inc.) with a resolution of 4 cm−1 was used at
the same angles of incidence. The measurements allowed for a
reliable determination of ε(ω) on a wavelength-by-wavelength
basis in the spectral range from ca. 0.15 eV up to 6 eV
(dots in Fig. 3).
In the spectral range 0.01–0.2 eV, the far infrared (FIR)
optical response of Zn-Mg-R quasicrystals was measured by
Fourier-transform (FT) reflectance spectroscopy. The measurements were carried out with a Vertex 70v vacuum FT
spectrometer with spectral resolution 4 cm−1 . The FIR reflectivity spectrum R(ω) of Zn-Mg-Y, appended in the IR-UV
region by SE data, is presented by dots in Fig. 4. The full curve
presents for a comparison data from Chernikov et al.15 As
seen, the present R(ω) spectrum is close to that of Chernikov
et al. in the long-wavelength region. However, it shows higher
reflectance in the VIS-UV range, which we suppose is due to
either different optical surface preparation techniques used
or different atomic compositions of investigated Zn-Mg-Y
samples.
The Zn-Mg-R dielectric function ε(ω) in FIR region
(presented by curves in Fig. 3) was determined from the
R(ω) spectrum with a Kramers-Kronig transform making use
of the anchor-window technique.24 The high-frequency R(ω)
asymptote (dashed curve in Fig. 4) was determined from the
requirement that the dielectric function determined by the
Kramers-Kronig transform would coincide with SE data in
IR-UV range.
3000
TABLE I. Physical parameters of fci Zn-Mg-R quasicrystals: the
mass density , the valence electron concentration n, and the Fermi
wave vector kF = (3π 2 n)1/3 .
Zn 62 Mg29 Y9
Zn65 Mg25 Ho10
Zn65 Mg24 Er11
(g/cm3 )
n
(10 cm−3 )
kF
(10 cm−1 )
5.18
5.82
5.87
1.17
1.13
1.12
1.51
1.50
1.49
23
2000
1000
dielectric function
40
dielectric function
30
200
fci Zn-Mg-Y
Im ( )
100
Re ( )
0
0.1
1
10
photon energy (eV)
0
0.01
0.1
1
10
photon energy (eV)
−8
FIG. 3. Dielectric function of Zn-Mg-Y quasicrystal. (Dots
present spectroscopic ellipsometry data, curves correspond to ε(ω)
determined by the Kramers-Kronig analysis of FIR reflectivity
spectrum.)
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PHYSICAL REVIEW B 88, 094201 (2013)
1.0
e 2 g 222001
g 222010
g
222100
0.8
reflectivity
e3
e1
e6
M
0.6
Z
P
e5
0.4
K
X
Y
Γ
fci Zn-Mg-Y
e4
0.2
(b)
(a)
0.0
0.01
0.1
1
FIG. 5. (Color online) Orientation of g 222100 , g 222001 , and g 222010
vectors (a) and the effective 222100 and 311111 Brillouin zone (b).
10
photon energy (eV)
FIG. 4. Reflectivity spectrum of the Zn-Mg-Y quasicrystal. (Dots
present our experimental data, full curve presents literature data,15
dashed curve corresponds to the high-frequency extrapolation used
in Kramers-Kronig analysis.)
III. MODEL OF ELECTRON ENERGY SPECTRUM
Since the Bloch theorem is not applicable for aperiodic
systems, the problem of an electronic structure of quasicrystals, despite of its importance, remains to be in principle
unsolved. The QC electronic structure is usually considered
in frameworks of two different approaches.
The first one treats the quasicrystalline potential acting on
the electron subsystem as that of the QC approximants. This
reduces the QC electronic structure problem to that of periodic
crystals with huge unit cells, containing large numbers of
atoms, and relevant techniques of band-structure calculations,
usually the tight-binding linear-muffin-tin-orbital (TB-LMTO)
method, are being employed to examine the QC valence bands
(see, e. g., reviews 25–27). The calculations predict a spiky
structure of the valence band density of states (DOS) and a
pseudogap feature at the Fermi level.
The alternative approach, the so-called band structure
hypothesis,28 treats the electronic subsystem in quasicrystals
as a NFE-gas, affected by a weak quasicrystalline potential
V (r) =
V g exp(i g · r).
(1)
g
(Here g are the reciprocal lattice vectors.) The approach,
when accounting for several Fourier amplitudes V g of the
QC potential only, yields as well a pseudogap structure, but
otherwise a smooth DOS dispersion.
Though the DOS spikiness is considered to be a distinctive
feature of quasicrystals, it was not observed experimentally.
This can be due to a finite broadening/scattering parameter =
h̄/τ of electron states. As has been shown by Fujiwara et al.,29
a value of = 0.7 eV completely washes out an anticipated
spiky structure of the optical conductivity spectrum. Since
the relaxation times in all known quasicrystals are short and
correspond to of the order of 0.1–1 eV, experimentally one
can not unambiguously distinguish between the two electron
energy spectrum models.
In our previous Zn-Mg-R PE studies,16–18 the bandstructure hypothesis approach was successfully applied for
an analysis of the valence-band PE spectra. The Zn-Mg-R
valence-band density of states was calculated in the framework
of the NFE approximation and the calculated DOS nicely
reproduced the observed PE valence-band spectra. Moreover,
the calculations16 yielded a correct experimental value of
the Zn-Mg-Y electronic specific-heat coefficient and agreed
well with the main Zn-Mg-Y DOS features, independently
calculated by the TB-LMTO technique.30,31
A. NFE approximation
The model of the Zn-Mg-R electronic structure, which was
used for an analysis of the PE data, is based on the NFE
approximation and treats the quasicrystalline potential (1) as
a small perturbation.
Though the reciprocal quasicrystalline lattice is dense, the
structure factors S g of most of its nodes are very low. The
g vectors, which play a decisive role in the quasicrystalline
potential (1), correspond to both high structure factors and
the Bragg planes, which are in the proximity of the Fermi
surface. This set of g vectors, which we will denote as G, can
be determined from the experimental diffraction patterns.
In Zn-Mg-R quasicrystals, G is comprised (see Fig. 1) of
twelve g 311111 vectors, which are directed along the C5 axes of
icosahedron, and sixty g 222100 vectors. The subset of g 222100
vectors is constituted of twenty triads, which group around C3
axes, as illustrated in Fig. 5(a) for the {g 222100 ,g 222001 ,g 222010 }
triad around g 222000 C3 . The 72 g ∈ G vectors determine
the effective Brillouin zone of Zn-Mg-R quasicrystals,16,17
the faces of which correspond to the g ∈ G Bragg planes
[Fig. 5(b)].
The electron energy spectrum problem can be simplified by
leaving only the g ∈ G terms in the Fourier expansion of the
quasicrystalline potential (1),
V (r) =
V g exp(i g · r),
(2)
g∈G
and in the expansion of the electron wave function
Ck−g k−g .
k = Ck k +
(3)
g∈G
Here k = −1/2 exp(ik · r) are the plane waves, and is
the normalization volume. The Fourier amplitude V0 of the
quasicrystalline potential (1) was eliminated by a shift of the
energy reference point.
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V. KARPUS et al.
PHYSICAL REVIEW B 88, 094201 (2013)
The standard procedure of a projection of the Schrödinger
equation onto the k and k−g eigenfunctions yields the
following system of equations for coefficients of the wavefunction expansion (3):
V g Ck−g = 0,
(4)
[k − ε(k)] Ck +
g∈G
[k−g − ε(k)]Ck−g + V−g Ck = 0.
(5)
Here ε(k) is the electron energy to be determined and
k = h̄2 k 2 /2m0 is the free electron energy. In a derivation of
Eqs. (4) and (5), it was taken into account that a difference of
any two g vectors of G, g 1 ∈ G and g 2 ∈ G, does not belong
to G, g 1 − g 2 ∈
/ G.
Equation (5) can be easily solved with respect to Ck−g ,
and by inserting the answer into (4) one obtains the dispersion
equation
2
V g ε(k) − k =
,
(6)
(ε(k) − k ) − (k−g − k )
g∈G
which determines the electron energy spectrum.
The dispersion equation (6) can be solved numerically
at given values of two parameters, the V222100 and V311111
Fourier amplitudes of the quasicrystalline potential (2). The
hypothetical Zn-Mg-R energy spectrum, calculated at the
V222100 = 0.31 eV and V311111 = 0.58 eV values determined
in an analysis of the experimental Zn-Mg-Y PE valence-band
spectra,16 is presented in Fig. 6. Since the V222100 and V311111
pseudopotentials acquire similar values in Zn-Mg-Y, Zn-MgHo, and Zn-Mg-Er quasicrystals,16,17 the energy spectrum
(Fig. 6) can be considered as a hypothetical band structure
in the vicinity of the Fermi level for all three, Zn-Mg-Y,
Zn-Mg-Ho, and Zn-Mg-Er, quasicrystals studied. The energy
spectrum in Fig. 6 is presented in ε0 = h̄2 (g222100 /2)2 /2m0
units and is depicted along the KXYPZMK path on faces of
the effective Brillouin zone, indicated in Fig. 5(b).
As seen in Fig. 6, the Fermi level is above the 222100 =
2|V222100 | pseudogap and lies in a lower part of the 311111 =
2|V311111 | pseudogap. The lower energy band is almost
1.3
1.1
k /
311111
F
222100
V 222100 = 0.31 eV
0.9
V 311111 = 0.58 eV
0.8
K
X
Y
P
Z
B. Model of independent intersections
The energy spectrum of the NFE gas is essentially affected
by the quasicrystalline potential at intersections of the isoenergetic surface with Bragg planes. In proximity to a chosen Bragg
plane, the dispersion equation (6) can be simplified—only a
single term, corresponding to the plane, is sufficient to retain
on its right-hand side,
δε g (k) =
|V g |2
.
δε g (k) − (k−g − k )
(7)
Here δε g (k) = ε(k) − k is the deviation from the free electron
spectrum due to an intersection with a single Bragg plane.
Solutions of Eq. (7) correspond to two well-known NFE
energy spectrum branches (see, e. g., Ref. 32), which in the
extended zone presentation acquire the form
g
γ 2
γ−
γ +1 .
(8)
δε g (k) =
2
|γ |
Here g = 2|V g | is the pseudogap, γ is a function of the
quasimomentum k component k along the g direction,
k−g − k
4ε0
k
=
1− 1
γ =
,
(9)
g
g
g
2
1.2
1.0
completely filled with electrons, the empty states, the hole
pockets, are left along lines connecting P and M points only.
The electrons fill up the higher energy band along the KY lines.
These features of the energy spectrum suggest a presumable
semimetallic behavior of the Zn-Mg-R quasicrystals. This is in
agreement with results of Hall effect measurements on Zn-MgY,15 which indicate that the effective electron concentration is
very low, neff = 2.3 × 1021 cm−3 , as compared to the valence
electron concentration n = 1.17 × 1023 cm−3 (Table I).
The Zn-Mg-R energy spectrum scheme presented in Fig. 6
allows for a prediction of the interband optical transitions
across 222100 and 311111 pseudogaps, which should result
in the optical feature at about 1 eV. The optical spectra,
in principle, can be numerically calculated making use of
the presented hypothetical band-structure scheme. However,
since only particular regions of k space (where the electron
energy spectrum deviates from the free-electron one) are of
importance for optical transitions, the analytical calculations,
which take into account a detailed description of the energy
spectrum in the vicinity of pseudogaps, are much more
effective.
M
K
wave vector
FIG. 6. (Color online) Energy spectrum of Zn-Mg-R quasicrystals in the vicinity of the Fermi level, at the pseudopotential values
V222100 = 0.31 eV, V311111 = 0.58 eV, and the Fermi energy εF =
9.31 eV determined from an analysis of the experimental Zn-Mg-Y
PE data.16
and ε0 = h̄2 ( 12 g)2 /2m0 is the intersection energy.
In the extended zone presentation, used in the present
study, the deviations δε g (k) are localized in close proximity
to Bragg planes, within the narrow strips the width of which,
when k is measured in 12 g units, is about g /4ε0 . Under
the assumption that strips do not overlap, the electron energy
spectrum, determined by Eq. (6), can be presented in the simple
form
δε g (k).
(10)
ε(k) = k +
g∈G
Within the model of independent intersections, the coefficients Ck−g of the wave-function expansion (3) are found from
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OPTICAL SPECTROSCOPY AND ELECTRONIC . . .
PHYSICAL REVIEW B 88, 094201 (2013)
(5) by replacing ε(k) → k + δε g (k),
Ck−g =
V−g
γ 2
γ−
γ + 1 Ck ,
|V g|
|γ |
A. Interband optical conductivity
(11)
whereas the Ck coefficient is found from the normalization
condition,
Within the framework of the independent intersections
model, the interband optical conductivity can be represented
by the following compact formula:
e2 g x1 dx S(x)
σib (ω) =
K(x,z,b),
(13)
√
24π h̄ 1 x 3 x 2 − 1
g∈G
⎡
⎤
2 −1
γ
|Ck |2 = ⎣1 +
γ−
γ2 + 1 ⎦ .
|γ
|
g∈G
K(x,z,b) =
(12)
The model of independent intersections breaks down at
edges and vertices of the effective Brillouin zone. Since they
have considerably smaller phase space than the BZ faces, the
model should be a relevant tool for a description of optical
transitions. However, one should keep in mind that an overlap
of the strips, within which the deviations δε g (k) are localized,
can become of importance at small angles between g vectors.
Then the model of independent intersections will overestimate
the partial contributions of Bragg planes. In the considered set
of g vectors, this can be the case for the (222100) triads. Indeed,
the angle between g vectors of the triad is comparatively small,
about 19.4◦ . To remedy the situation, when modeling an optical
response of Zn-Mg-R quasicrystals, we will treat each triad
of the (222100) subset as a single Bragg plane, i. e., we will
replace the subset of sixty g 222100 vectors by twenty effective
g vectors, moduli of which are equal to that of g 222100 .
IV. OPTICAL CONDUCTIVITY
The optical conductivity in metallic and semimetallic solids
is determined by the intraband, Drude-type, and the interband
transitions of the valence electrons. Both terms of the optical
conductivity, σDrude (ω) and σib (ω), can be calculated following
the Ashcroft and Sturm algorithm,21 which was worked out
for the usual crystalline metals, and later was successfully
applied to an analysis of the fci Al-Cu-Fe optical response.33
The Ashcroft and Sturm calculations were carried out in the
reduced zone presentation. Since the reduction of electron
states to the Brillouin zone is based on the Bloch theorem
and, therefore, can not be strictly justified for quasicrystals, we
repeated optical conductivity calculations in the extended zone
scheme. The main difference in the reduced- and extendedzone presentations is the selection rule for optical transitions.
Within the reduced zone scheme, the transitions are vertical,
k = k, whereas within the extended zone presentation the
transitions are indirect—the final quasimomentum is shifted
by the reciprocal lattice vector, k = k − g. Nevertheless, as
can be expected, calculations in both presentations lead to the
same final results for σ (ω).
The Ashcroft and Sturm σ (ω) formulas were derived for
polyvalent metals, the Fermi level in which is positioned above
a pseudogap. Here we will supplement the Ashcroft and Sturm
results by formulas appropriate to the cases when the Fermi
level is positioned within a pseudogap and below it. The final
σib (ω) and σDrude (ω) formulas are presented in the following
subsections, and a scheme of their derivation is outlined in
Appendix.
z
iπ
1
1
+
.
x − (z + ib) x + (z + ib)
(14)
Here the dimensionless integration variable x is related to γ
in Eq. (9) as x = (γ 2 + 1)1/2 . The dimensionless parameters
z = h̄ω/ g and b = / g of the kernel K(x,z,b) correspond
to the photon energy h̄ω and the broadening parameter ,
respectively. The S(x) function acquires different expressions
depending on the relative position of the Fermi level with
respect to a pseudogap:
(i) The Fermi level below a pseudogap,
0,
1 < x < x0 ,
S(x) = (x−x0 )(x1 −x)
(15)
,
x0 < x < x1 .
2(x1 +x0 )
(ii) The Fermi level within a pseudogap,
S(x) =
(x − x0 )(x1 − x)
, 1 < x < x1 .
2(x1 + x0 )
(iii) The Fermi level above a pseudogap,
x,
1 < x < |x0 |,
S(x) = (x−x0 )(x1 −x)
, |x0 | < x < x1 .
2(x1 +x0 )
(16)
(17)
Here the dimensionless energies x0 and x1 are defined as
⎡
⎤
2
g ⎦
εF
4ε0 ⎣
1−
,
(18)
x0 =
+
g
ε0
4ε0
⎡
⎤
g 2⎦
εF
4ε0 ⎣
x1 =
+
1+
.
g
ε0
4ε0
(19)
B. Intraband optical conductivity
The intraband optical conductivity has the Drude-type
dispersion
σdc
,
(20)
σDrude (ω) =
1 − iωτ
where σdc is the static electric conductivity and τ is the
relaxation time.
The deviations of the electron energy spectrum from the
free electron parabola affect the group velocity of electrons
and, consequently, reduce the static conductivity. The effect is
accounted for by the optical mass mopt , which enters the static
conductivity formula σdc = e2 nτ/mopt . Within the model of
independent intersections, the optical mass is given by the
formula
⎡
⎤
m0 g g
1 ⎣
1
1−
=
C⎦ ,
(21)
2
mopt
m0
24π
h̄
n
g∈G
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V. KARPUS et al.
PHYSICAL REVIEW B 88, 094201 (2013)
of optical conductivity to satisfy the oscillator sum rule,
∞
ωp2
dω Re σ (ω) =
.
(26)
W =
8
0
1.0
C
g
= 0.05
0
C. Limiting cases
0.5
g
= 0.1
The integrals in formulas (13) and (23) of the interband
conductivity and the C coefficient can be calculated analytically. However, the resulting algebraic expressions are lengthy
and complicated, and we will not present them here.
The transparent formulas for the interband conductivity can
be obtained in the limiting cases of weak and strong scattering.
0
0.0
0.8
0.9
1.0
F
/
1.1
1. Weak scattering limit
1.2
0
FIG. 7. Dependence of the C coefficient on the position of the
Fermi level with respect to a pseudogap (ε0 is the intersection energy,
ε0 = h̄2 (g/2)2 /2m0 ).
and the intraband optical conductivity can be represented in
the form
⎡
⎤
2
2
τ
ω
e g τ g ⎦
1
⎣ p −
σDrude (ω) =
C , (22)
1 − iωτ 4π
24π
h̄ h̄
g∈G
where ωp = 4π e2 n/m0 is the plasma frequency. The dimensionless coefficient C in formulas (21) and (22) is defined by
the expression
2 x1 dx S(x)
,
(23)
C=
√
π 1 x2 x2 − 1
where S(x) function is determined by (15)–(17).
The C coefficient depends on the position of the Fermi
level with respect to a pseudogap and on a pseudogap value.
The C = C(εF /ε0 ) function at several given g /ε0 values is
presented graphically in Fig. 7. As seen, a contribution of
pseudogaps to the intraband conductivity decreases when they
are raised above the Fermi level.
The C coefficient determines the dependence of both
intraband and interband optical conductivity on the position
of the Fermi level with respect to a pseudogap.
∞ Indeed, the
spectral weight of the interband transitions, 0 dω Reσib (ω),
as can be easily proved by a straightforward integration of the
K-kernel (14), is given by the formula
∞
e2 g g
dω Re σib (ω) =
C.
(24)
Wib =
48h̄2
0
g∈G
As it can be seen, the intensity of the interband absorption
peaks is proportional to the pseudogap width g and to the C
coefficient.
The spectral weight of the Drude peak is
∞
ωp2 e2 g g
WDrude =
−
dω Re σDrude (ω) =
C. (25)
8
48h̄2
0
g∈G
As seen from (24) and (25), the spectral weight of the interband
absorption is compensated by a reduction of the Drude peak by
exactly the same amount. This ensures the total spectral weight
In the limit of weak scattering, when g , the K kernel
(14) reduces to the form
z 1
1
+ iπ δ(x − z) + x+z
+ iπ δ(x + z) ,
K(x,z,b) =
x−z
iπ
(27)
and the real part of the interband conductivity is easily
calculated due to a simple integration with the δ(x − z)
function. The interband absorption line/band due to a single
Bragg plane acquires the form
g 2 S(x)|x=h̄ω/ g
e2 g
g
.
(28)
Re σib (ω) =
24π h̄ h̄ω
(h̄ω/ g )2 − 1
When the Fermi level is below a pseudogap, the S(x)
function is given by (15), and formula (28) corresponds to
the well-known Butcher result,34 widely used in an analysis
of the optical conductivity of alkali metals. When the Fermi
level is above a pseudogap, the S(x) function is given by
(17), and formula (28) repeats the Ashcroft and Sturm21 and
Golovashkin et al.35 results, derived for polyvalent metals.
When the Fermi level is positioned within a pseudogap,
ε0 − 12 g < εF < ε0 + 12 g , the S(x) function is given by
(16), and formula (28) predicts an interband absorption band
with the absorption edge determined by the pseudogap
width
g
g . The absorption onset has the Re σib (ω) ∝ 1/ h̄ω − g
singularity, which is smeared out at finite values of the
broadening parameter (Fig. 8).
2. Strong scattering limit
In the limit of strong scattering (considered by Burkov
et al.),33 when g , the K kernel (14) reduces to the form
K(x,z,b) =
i2xz
.
π (z + ib)2
(29)
The interband optical conductivity due to a single Bragg plane
in this limit is given by the formula
g
σib (ω) =
ih̄ω
e2 g g
C
.
24π h̄ (h̄ω + i)2
(30)
The position of the absorption band is now determined by the
broadening parameter .
The strong scattering limit, g , is relevant for very
narrow pseudogaps, which in quasicrystals correspond to the
reciprocal lattice nodes with low structure factors S g . When
modeling optical conductivity of Zn-Mg-R quasicrystals, we
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PHYSICAL REVIEW B 88, 094201 (2013)
= 0
g = 0.1
optical conductivity (Ω-1cm-1)
1.5
= 0.05
0
g
1.0
)/
0
F
= 0.2
0.5
0.0
0.0
1.0
h- /
g
g
2.0
g
FIG. 8. The interband absorption band when the Fermi level is
positioned within a pseudogap. Full curve corresponds to Eq. (28),
while dashed curves [Eq. (13)] show the band at finite values of the
broadening parameter . The optical conductivity is presented in
σ0 = e2 g/24π h̄ units.
will use formula (30) to account for a possible contribution
of the low-S g pseudopotentials, which correspond to g ∈
/G
vectors. Another anticipated effect of the low-S g pseudopotentials on optical spectra, which was not dealt with in the
present study, is a broadening of the main, G-induced, optical
features by the low-S g scattering.
optical conductivity (Ω-1cm-1)
The experimental optical conductivity σ (ω) spectra of ZnMg-R quasicrystals are presented by dots in Fig. 9. All three,
Zn-Mg-Y, Zn-Mg-Ho, and Zn-Mg-Er, quasicrystals manifest
a similar dispersion—the spectra exhibit a distinct optical
feature at about 1 eV, which obviously is due to the interband
transitions, and the intraband Drude-type contribution at low
frequencies. A small kink in the Zn-Mg-Ho and Zn-Mg-Er
spectra at about 0.13 eV is an artefact. It corresponds to
a division line between data recorded by FIR reflection
spectroscopy and IR spectroscopic ellipsometry techniques.
The theoretical simulations of the σ (ω) spectra, based on
the formulas presented in the previous section, can easily
be carried out at given values of the Fermi energy εF , the
pseudogap widths 222100 and 311111 , and the broadening
parameters.
Although the dominant role in an optical response of
Zn-Mg-R, as well as of other metals, is played by the valence
electrons, one can expect a small possible contribution of
the bound Zn 3d core electrons. Since the Zn 3d level is at
the bottom of the valence band, approximately 10 eV below the
Fermi level,16,17 a feasible Zn 3d polarization was taken into
account by introducing the high-frequency dielectric constant
into the QC dielectric function
4π
[σDrude (ω) + σib (ω)] .
ω
(31)
Results of the Zn-Mg-R optical conductivity calculations
are presented by curves in Fig. 9. As seen, the theory nicely
reproduces the experimental data.
Re σ(ω )
6000
4000
fci Zn-Mg-Y
2000
Im σ(ω)
0
8000
0.1
(b)
1
10
photon energy (eV)
Re σ(ω)
6000
4000
fci Zn-Mg-Ho
2000
Im σ(ω )
0
-2000
0.01
V. ANALYSIS OF Zn-Mg-R OPTICAL SPECTRA
ε(ω) = ε∞ + i
(a)
0.01
3.0
optical conductivity (Ω-1cm-1)
Re
g
ib(
= 0.1
8000
8000
0.1
(c)
1
10
photon energy (eV)
Re σ(ω)
6000
4000
fci Zn-Mg-Er
2000
Im σ(ω)
0
-2000
0.01
0.1
1
10
photon energy (eV)
FIG. 9. (Color online) The optical conductivity spectra of (a) ZnMg-Y, (b) Zn-Mg-Ho, and (c) Zn-Mg-Er quasicrystals. Dots present
experimental data. Curves correspond to theoretical calculations.
The high-frequency dielectric constant, which was treated
as a fitting parameter, is, as expected, close to unity, ε∞ =
1.35, 1.56, and 1.58 for Zn-Mg-Y, Zn-Mg-Ho, and Zn-MgEr, respectively, and does not essentially influence an optical
response of the quasicrystals.
The plasma frequency ωp values, which determine the
total spectral weight of Re σ (ω) spectra [see Eq. (26)], are
of about 11.5–11.9 eV (Table II) and are close to the theoZn−Mg−Y
Zn−Mg−Ho
retical values ωp
= 12.7 eV, ωp
= 12.5 eV,
Zn−Mg−Er
= 12.4 eV, predicted by the formula ωp =
and
ωp
4π e2 n/m0 at the valence electron concentration n (Table I),
which was determined from the mass density and the average
valence values Z Zn−Mg−Y = 2.09, Z Zn−Mg−Ho = 2.10, and
Z Zn−Mg−Er = 2.11.
Figure 10 shows the intraband and interband contributions,
σDrude (ω) and σib (ω), to the Zn-Mg-Y total optical conductivity
094201-7
V. KARPUS et al.
PHYSICAL REVIEW B 88, 094201 (2013)
TABLE II. Parameters of the Zn-Mg-R intraband Drude optical
conductivity.
11.5
6100
0.18
0.06
16
Zn-Mg-Er
11.9
5200
0.45
0.12
8.1
( ) (Ω-1cm-1)
ωp (eV)
σdc (
−1 cm−1 )
h̄/τ (eV)
wDrude
mopt (m0 )
Zn-Mg-Ho
6000
11.7
5500
0.47
0.14
7.2
(a)
fci Zn-Mg-Y
4000
Re σib( )
2000
ib
Zn-Mg-Y
8000
0
Im σib( )
-2000
-4000
0.01
spectrum. The relative spectral weight of the intraband Drude
conductivity, wDrude = WDrude /W , with respect to the total
σ (ω) spectral weight W , is of about 10% (see Table II). The
relative spectral weight wDrude , as can easily be checked from
formulas (21) and (25), unambiguously determines the optical
mass, mopt = m0 /wDrude , which, therefore, in the fci Zn-Mg-R
quasicrystals is approximately 10 m0 .
The parameters of the intraband optical conductivity, the
static conductivity and the relaxation time, are presented in
Table II. The determined static conductivity values σdc ∼
5000–6000 −1 cm−1 roughly correspond to the Fisher et al.
results of the electric resistivity measurements:36,37 σdc ≈
6620, 5410, and 5880 −1 cm−1 for Zn-Mg-Y, Zn-MgHo, and Zn-Mg-Er, respectively. The relaxation times τ ∼
0.14–0.4 10−14 s determined in the present study are longer
than the τ values we reported previously,20 where they were
deduced from the σ (ω) analysis in a narrower spectral range of
Re ( ) (Ω-1cm-1)
4000
( ) (Ω-1cm-1)
ib
0
Im σib( )
( ) (Ω-1cm-1)
0.1
10
1
photon energy (eV)
(c)
fci Zn-Mg-Er
4000
2000
Re σib( )
0
Im σib( )
-2000
-4000
0.01
0.1
1
10
photon energy (eV)
FIG. 11. (Color online) Interband optical conductivity of (a) ZnMg-Y, (b) Zn-Mg-Ho, and (c) Zn-Mg-Er quasicrystals.
0.1
10
1
photon energy (eV)
Im σDrude( )
2000
0
-2000
Im σib( )
-4000
0.01
Re σib( )
2000
6000
Re σib( )
0
60000.01
fci Zn-Mg-Ho
4000
8000
Re σDrude( )
10
1
(b)
-4000
0.01
6000
4000
0.1
photon energy (eV)
-2000
(a)
fci Zn-Mg-Y
2000
-1
-1
Im ( ) (Ω cm )
6000
ib
8000
8000
0.1
1
(b)
10
photon energy (eV)
FIG. 10. (Color online) The intraband and interband contributions, σDrude (ω) and σib (ω), to the Zn-Mg-Y total optical conductivity
spectrum.
0.1–6 eV, but closer to the Zn-Mg-Y and Zn-Mg-Tb relaxation
times reported by Chernikov et al.,15 τ ∼ 0.5 10−14 s.
The Zn-Mg-R interband optical conductivity σib (ω) spectra,
which were obtained by a subtraction of the determined Drude
contribution σDrude (ω) from the total optical conductivity, are
presented in Fig. 11, where dots and curves correspond to
experimental data and theoretical calculations, respectively.
The interband optical conductivity is due to the optical
transitions across the 222100 and 311111 pseudogaps and due
to a contribution of the g ∈
/ G pseudopotentials with low structure factors. The relative spectral weights of Re σib,222100 (ω),
Re σib,311111 (ω), and Re σib,g∈G
/ (ω) interband conductivities
(with respect to the total spectral weight W ) are around 0.5, 0.3,
and 0.1, respectively (see Table III). The partial contributions
to the Zn-Mg-Y interband optical conductivity spectrum are
shown in Fig. 12.
The parameters of the electron energy spectrum, the
Fermi energy εF , the pseudogap widths 222100 and 311111 ,
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PHYSICAL REVIEW B 88, 094201 (2013)
TABLE III. The relative spectral weights of the interband transitions across the 222100 and 311111 pseudogaps and of the transitions
induced by the low S g -factor pseudopotentials.
Zn-Mg-Y
Zn-Mg-Ho
Zn-Mg-Er
0.51
0.29
0.14
0.50
0.29
0.08
0.49
0.25
0.12
w222100
w311111
wg∈G
/
and the broadening parameters, which were determined by
standard least-squares technique, are presented in Table IV.
The Fermi energy εF = 9.20–9.25 eV and the pseudogap
222100 = 0.55–0.61 eV and 311111 = 1.34–1.45 eV values
are rather close to those, deduced from an analysis of the
Zn-Mg-R valence band photoemission spectra,16,17 namely,
9.31–9.32 eV for the Fermi energy, and 0.61–0.64 eV and
0.98–1.25 eV for the 222100 and 311111 pseudogaps. This is a
sound support for a self-consistency of the QC electron energy
spectrum model suggested—it nicely reproduces experimental
data for both the optical and the PE response of the Zn-Mg-R
quasicrystals at actually the same values of energy spectrum
parameters.
The broadening parameter G of the interband transitions
across the pseudogaps, which was assumed to be the same
for 222100 and 311111 , is of about 0.62–0.68 eV. It differs
from the broadening parameter determined in PE analysis,
PE ≈ 0.22–0.24 eV. We do not know the exact reasons
for the difference. Here it should be mentioned that the
TABLE IV. Parameters of fci Zn-Mg-R electron energy spectrum:
the Fermi energy εF , the pseudogaps 222100 and 311111 , and the
broadening parameters 222100 = 311111 ≡ G and g∈G
/ . The values
of the parameters, deduced from an analysis of the photoemission
data,16,17 are presented in parentheses.
εF (eV)
222100 (eV)
311111 (eV)
G (eV)
g∈G
/ (eV)
g∈G
/ (eV)
Zn-Mg-Y
Zn-Mg-Ho
Zn-Mg-Er
9.25 (9.31)
0.55 (0.63)
1.36 (1.16)
0.62 (0.23)
g∈G
/
0.35
9.25 (9.32)
0.60 (0.61)
1.45 (0.98)
0.63 (0.22)
g∈G
/
0.41
9.20 (9.32)
0.61 (0.64)
1.34 (1.25)
0.68 (0.24)
g∈G
/
0.55
broadening in the optical and PE data analysis is introduced
by different phenomenological schemes: in optics, as a
Lorentzian-type broadening of the spectral lines [see the
K-kernel formula (14)], and in PE, as a Gaussian convolution
of the density of states.
The broadening parameter g∈G
/ of the interband transitions
induced by the pseudopotentials with low structure factors
is about 0.35–0.55 eV (Table IV). Since the line shape of
σib,g∈G
/ (ω) is determined solely by the broadening parameter
[see Eq. (30)], the pseudopotentials Vg∈G
/ and the corresponding pseudogaps g∈G
/ = 2|Vg ∈G
/ | can not be directly deduced
from the optical spectra analysis. They determine the spectral
weight of the {g ∈
/ G}-transitions,
Wg∈G
/ =
-1
-1
) (Ω cm )
8000
ib(
Re
g ∈G
/
(a)
6000
Re σib( )
Re σib,222100( )
4000
Re σib, g
2000
G
( )
Re σib,311111( )
0
4000
) (Ω-1cm-1)
ib(
Im
Im σib( )
(b)
fci Zn-Mg-Y
2000
Im σib, g
G
( )
0
Im σib,311111( )
-2000
Im σib,222100( )
-4000
0.01
e2 gg∈G
/
0.1
1
10
photon energy (eV)
FIG. 12. (Color online) The partial contributions σib,222100 (ω),
σib,311111 (ω), and σib,g∈G
/ (ω) to the Zn-Mg-Y interband optical conductivity spectrum.
48h̄2
C.
(32)
A rough estimate of the g∈G
/ pseudogaps can be obtained
assuming that all g vectors contributing to the sum in
Eq. (32) have the same moduli, g ≈ 2kF , and correspond to
the same pseudogap values. Then, under assumption that a
multiplicity of the g ∈
/ G vectors is 60 and that C ≈ 1, we
obtain an estimate of g∈G
/ ∼ 0.03–0.04 eV. Such low values
of g∈G
/ pseudogaps seem to be physically acceptable taking
into account that, in the Zn-Mg-R high-resolution diffraction
patterns, the Bragg peaks with their intensity ratios below
∼10−3 manifest themselves.38
Finally we would like to point out that, due to a rather
smooth spectral shape of the experimental Zn-Mg-R optical
spectra, a determination of both intraband, and interband
parameters, most probably, is not unambiguous. Previously
we qualitatively explained the experimental Zn-Mg-R optical
conductivity spectra, recorded in the narrower spectral range
of 0.1–6 eV, by optical transitions across a single 311111
pseudogap.20 However, this required very short Drude relaxation times, τ ∼ 0.03 10−14 s (h̄/τ ∼ 2 eV), a decrease
of the 311111 value down to ∼ 0.8 eV, and an increase of
the Fermi energy up to 9.7–9.8 eV. The reliability of the set
of parameters, determined in the present study, we suppose,
is confirmed by the fact that the electron energy spectrum
in the vicinity of the Fermi level is predicted to have the
same structure as that predicted from the photoemission data
analysis (Fig. 13)—the Fermi level is slightly above the 222100
pseudogap, and is positioned in a lower part of the 311111
pseudogap.
094201-9
V. KARPUS et al.
PHYSICAL REVIEW B 88, 094201 (2013)
311111
energy (eV)
10
isotropic system has the form
σib (ω) =
F
9
222100
where fk is the Fermi distribution function and fk k is the
oscillator strength
222100
(a)
(A1)
k= k
311111
F
f k f k k
2e2h̄2 ω ,
i3m0 [ε(k ) − ε(k)]2 − (h̄ω + i)2
(b)
f k k =
FIG. 13. The Zn-Mg-Y electron energy spectrum in the vicinity
of the Fermi level, predicted from analysis of optical (a) and
photoemission spectroscopy (b) data.
VI. SUMMARY
Summarizing, we presented the Zn-Mg-R (R = Y, Ho, Er)
optical conductivity spectra, recorded in the 0.01–6 eV spectral
range by a combined spectroscopic ellipsometry and FIR
reflection spectroscopy technique. The model of Zn-Mg-R
electron energy spectrum, suggested previously in an analysis
of the valence-band photoemission spectra,16 was developed—
a hypothetical band structure of the quasicrystals in the vicinity
of the Fermi level was calculated and the NFE model of
independent intersections was formulated in the extended zone
presentation. The Ashcroft and Sturm scheme21 of the optical
conductivity theoretical calculations was extended to account
for various positions of the Fermi level with respect to a
pseudogap.
An analysis of the Zn-Mg-R optical conductivity spectra
shows that the intraband, Drude-type, optical transitions
contribute to the total optical conductivity with a relative
spectral weight of about 10%. The Drude relaxation times
are about 0.14–0.4 10−14 s.
The interband optical conductivity is predominantly due to
the optical transitions across 222100 and 311111 pseudogaps.
Their relative spectral weight is about 80%. The relative
spectral weight of an additional contribution due to the
pseudopotentials, corresponding to the reciprocal lattice nodes
with low structure factors, is about 10%.
The experimental Zn-Mg-R optical conductivity spectra
are reproduced in detail by theoretical calculations performed
within the framework of the suggested electron energy spectrum model. The set of electron energy spectrum parameters
determined from an analysis of the optical data predicts actually the same Fermi-level vicinity electron energy spectrum
structure as was previously predicted from an analysis of
photoemission data.16,17
2
m0
[ε(k )
− ε(k)]
|k | p|k|2 .
(A2)
The straightforward calculations of the oscillator strength
within the model of independent intersections, when the
electron wavefunctions and energy spectrum are given by
formulas (3) and (10)–(12), yield the following answer for
the oscillator strength:
f k k =
γ
4ε0
δ
,
2 + 1)3/2 k ,k−g
|γ
|
(γ
g
g∈G
(A3)
where ε0 = h̄2 ( 21 g)2 /2m0 is the intersection energy, g is the
pseudogap width, and γ is the dimensionless function (9) of
the quasimomentum k component k along the g-direction.
Inserting (A3) into (A1), taking a sum over the final states
k , and, changing a sum over k to an integral, one obtains the
intermediate σib (ω) expression
e2h̄2 ω 4ε0
γ
σib (ω) =
d 3k
3m
2 + 1)3/2
i12π
|γ
|
(γ
0
g
g∈G
×
( g
fk
γ2
+ 1)2 − (h̄ω + i)2
(A4)
.
Choosing the cylindrical coordinate system for kintegration with a polar axis directed along g-vector, the
formula (A4) can be rewritten as
e2 ω ∞
4ε0
σib (ω) =
dk
2
i6π −∞
g (1 + γ 2 )3/2
g∈G
×
S(k )
( g 1 + γ 2 )2 − (h̄ω + i)2
,
(A5)
where S(k ) is the dimensionless function defined as
S(k ) =
γ h̄2
|γ | m0
∞
k⊥ dk⊥ fk .
(A6)
0
ACKNOWLEDGMENTS
The authors gratefully acknowledge support by the Research Council of Lithuania (Grant No. MIP-081/2012). The
Knut and Alice Wallenberg foundation is acknowledged for
support to instrumentation.
APPENDIX: DERIVATION OF σ (ω) FORMULAS
1. Interband optical conductivity
In the extended zone presentation, the general formula
of the interband optical conductivity39,40 for the optically
As seen from (A5), the interband optical conductivity is a
sum of partial contributions of the g ∈ G Bragg planes. Each
term of the sum is localized in the k-space in a vicinity of
the corresponding Bragg plane. Therefore, when calculating
the S(k )-function, the electron energy ε(k), which enters the
Fermi distribution, can be replaced by ε(k) = k +
√δε g (k).
The final answers for S(k ), as a function of the x ≡ γ 2 + 1
argument, are presented by formulas√(15)–(17). Replacing in
(A5) the integration variable to x = γ 2 + 1, one reduces the
σib (ω) formula to the final form (13).
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PHYSICAL REVIEW B 88, 094201 (2013)
2. Intraband optical conductivity
The general formula of the intraband optical
conductivity39,40 for the optically isotropic system has
the form
σDrude (ω) =
σdc
e2 nτ
, σdc =
,
1 − iωτ
mopt
(A7)
where the optical mass in the extended zone presentation is
determined by the formula
1
1
1
=
(A8)
d 3 k fk 2 ∇k2 ε(k).
3
mopt
4π n
3h̄
Here ∇k = ∂/∂ k is the gradient operator with respect to k.
Action of the ∇k2 operator on the energy spectrum (10)
yields
4ε0
1 2
1
1 γ
∇
ε(k)
=
−
. (A9)
k
2
2 + 1)3/2
m
3m
|γ
|
(γ
3h̄
0
0 g∈G
g
*
Corresponding author: [email protected]
J.-M. Frigerio and J. Rivory, J. Non-Cryst. Solids 117/118, 812
(1990).
2
C. C. Homes, T. Timusk, X. Wu, Z. Altounian, A. Sahnoune, and
J. O. Ström-Olsen, Phys. Rev. Lett. 67, 2694 (1991).
3
V. Demange, A. Milandri, M. C. de Weerd, F. Machizaud, G.
Jeandel, and J. M. Dubois, Phys. Rev. B 65, 144205 (2002).
4
L. Degiorgi, M. A. Chernikov, C. Beeli, and H. R. Ott, Solid State
Commun. 87, 721 (1993).
5
M. A. Chernikov, L. Degiorgi, A. Bernasconi, C. Beeli, and H. R.
Ott, Physica B 194–196, 405 (1994).
6
D. N. Basov, F. S. Pierce, P. Volkov, S. J. Poon, and T. Timusk,
Phys. Rev. Lett. 73, 1865 (1994).
7
H. Werheit, R. Schmechel, K. Kimura, R. Tamura, and
T. Lundström, Solid State Commun. 97, 103 (1996).
8
A. D. Bianchi, F. Bommeli, M. A. Chernikov, U. Gubler, L.
Degiorgi, and H. R. Ott, Phys. Rev. B 55, 5730 (1997).
9
X. Wu, C. C. Homes, S. E. Burkov, T. Timusk, F. S. Pierce, S. J.
Poon, S. L. Cooper, and M. A. Karlow, J. Phys.: Condens. Matter
5, 5975 (1993).
10
D. N. Basov, T. Timusk, F. Barakat, J. Greedan, and B. Grushko,
Phys. Rev. Lett. 72, 1937 (1994).
11
A. D. Bianchi, F. Bommeli, E. Felder, M. Kenzelmann, M. A.
Chernikov, L. Degiorgi, H. R. Ott, and K. Edagawa, Phys. Rev. B
58, 3046 (1998).
12
V. Karpus, G.-J. Babonas, A. Rėza, W. Assmus, and R. Sterzel,
Lith. J. Phys. 40, 118 (2000).
13
A. Suchodolskis, W. Assmus, G.-J. Babonas, L. Giovanelli, U. O.
Karlsson, V. Karpus, G. Le Lay, A. Rėza, and E. Uhrig, Acta Phys.
Pol. A 107, 412 (2005).
14
V. Karpus, G.-J. Babonas, A. Rėza, A. Suchodolskis, S. Tumėnas,
O. Hunderi, W. Assmus, and S. Brühne, Acta Phys. Pol. A 113,
1005 (2008).
15
M. A. Chernikov, S. Paschen, E. Felder, P. Vorburger, B. Ruzicka,
L. Degiorgi, H. R. Ott, I. R. Fisher, and P. C. Canfield, Phys. Rev.
B 62, 262 (2000).
1
Inserting (A9) into (A8), one obtains the intermediate
expression
⎡
⎤
4ε
f
1 1 ⎣
γ
1
0 k
⎦.
1−
=
d 3k
mopt
m0
12π 3 n g∈G
|γ | g (γ 2 + 1)3/2
(A10)
Choosing the cylindrical coordinate system for k-integration,
the formula (A10) can be rewritten as
⎡
⎤
1
4ε0 S(k ) ⎦
1 ⎣
m0 ∞
=
dk
1−
,
mopt
m0
g (γ 2 + 1)3/2
6π 2 nh̄2 g∈G −∞
(A11)
where S(k ) function is defined by (A6).
√ Replacing in (A11) the k -integration variable to x =
γ 2 + 1, one reduces the optical mass formula to its final
form (21).
16
A. Suchodolskis, W. Assmus, L. Giovanelli, U. O. Karlsson,
V. Karpus, G. Le Lay, R. Sterzel, and E. Uhrig, Phys. Rev. B 68,
054207 (2003).
17
A. Suchodolskis, W. Assmus, L. Giovanelli, U. O. Karlsson,
V. Karpus, G. Le Lay, and E. Uhrig, J. Phys.: Condens. Matter
16, 9137 (2004).
18
V. Karpus, A. Suchodolskis, J. Taulavičius, U. O. Karlsson,
W. Assmus, S. Brühne, and E. Uhrig, Opt. Mater. 30, 690 (2008).
19
V. Karpus, G.-J. Babonas, A. Rėza, S. Tumėnas, H. Arwin,
W. Assmus, and S. Brühne, Z. Kristallogr. 224, 39 (2009).
20
S. Tumėnas, V. Karpus, H. Arwin, and W. Assmus, Thin Solid Films
519, 2951 (2011).
21
N. W. Ashcroft and K. Sturm, Phys. Rev. B 3, 1898 (1971).
22
A. Langsdorf and W. Assmus, J. Cryst. Growth 192, 152 (1998).
23
A. Langsdorf and W. Assmus, Cryst. Res. Technol. 34, 261 (1999).
24
S. Tumėnas, I. Kašalynas, V. Karpus, and H. Arwin, Acta Phys. Pol.
A 119, 140 (2011).
25
T. Fujiwara, in Physical Properties of Quasicrystals, edited by
Z. M. Stadnik (Springer, Berlin, 1999), pp. 169–207.
26
J. Hafner and M. Krajčı́, in Physical Properties of Quasicrystals,
edited by Z. M. Stadnik (Springer, Berlin, 1999), pp. 209–256.
27
Y. Ishii and T. Fujiwara, in Quasicrystals, edited T. Fujiwara and
Y. Ishii (Elsevier, The Netherlands, 2008), pp. 171–208.
28
A. P. Smith and N. W. Ashcroft, Phys. Rev. Lett. 59, 1365 (1987).
29
T. Fujiwara, S. Yamamoto, and G. Trambly de Laissardière,
Phys. Rev. Lett. 71, 4166 (1993).
30
M. Krajčı́ and J. Hafner, J. Phys.: Condens. Matter 12, 5831 (2000).
31
K. Oshio and Y. Ishii, J. Alloys Compounds 341, 402 (2002).
32
N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt,
Rinehart and Winston, New York, 1976).
33
S. E. Burkov, T. Timusk, and N. W. Ashcroft, J. Phys.: Condens.
Matter 4, 9447 (1992).
34
P. N. Butcher, Proc. Phys. Soc. (London) A 64, 756 (1951).
35
A. I. Golovashkin, A. I. Kopeliovich, and G. P. Motulevich, Zh.
Eksp. Teor. Fiz. 53, 2053 (1967) [Sov. Phys. JETP 26, 1161
(1968)].
094201-11
V. KARPUS et al.
36
PHYSICAL REVIEW B 88, 094201 (2013)
I. R. Fisher, Z. Islam, A. F. Panchula, K. O. Cheon, M. J.
Kramer, P. C. Canfield, and A. I. Goldman, Phil. Mag. B 77, 1601
(1998).
37
I. R. Fisher, K. O. Cheon, A. F. Panchula, P. C. Canfield,
M. Chernikov, H. R. Ott, and K. Dennis, Phys. Rev. B 59, 308
(1999).
38
A. Létoublon, I. R. Fisher, T. J. Sato, M. de Boissieu, M. Boudard,
S. Agliozzo, L. Mancini, J. Gastaldi, P. C. Canfield, A. I. Goldman,
and A.-P. Tsai, Mater. Sci. Eng. A 294–296, 127 (2000).
39
H. Ehrenreich and H. R. Philipp, Phys. Rev. 128, 1622 (1962).
40
H. Ehrenreich, H. R. Philipp, and B. Segall, Phys. Rev. 132, 1918
(1963).
094201-12